These are lyrics from "Memories", a song by an American pop-rock band called Maroon 5. The differences:
'Here's to...' is a conventional way of proposing a toast (an expression of good wishes or respect for someone that involves holding up and then drinking from a glass of alcohol). Example: Here's to Grandfather, on the occasion of his 90th birthday! (everyone raises their glass and typically says loudly, '[to] Grandfather!').
'Cheers!' is normally said alone and is a friendly expression said just before you drink an alcoholic drink. Example: some people are about to start drinking together. One says 'Cheers!' (just that) and the others repeat 'Cheers!' and may raise their glasses or just pick them up normally and start to drink from them.
Although dictionaries generally define toasts in connection with alcoholic beverages, people drinking other types e.g. juice, or even water, can propose or participate in toasts.
In prose, poetry and song lyrics, 'here's to...[someone or something]' can be used figuratively to compactly express support or admiration. An actual alcoholic toast may not be implied.
It is unusual to say 'Cheers to...' as the commencement of a toast, and I suspect that the words were used in that place, as so often in song lyrics, for reasons of rhyme, metre, etc.